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1. Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems

   https://doi.org/10.1109/TRO.2021.3076581  

   Mamakoukas, Giorgos; L. Castano, Maria; Tan, Xiaobo; D. Murphey, Todd (May 2021, IEEE Transactions on Robotics)

   null (Ed.)

   This paper presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error in terms of the prediction horizon and the magnitude of the derivatives of the system states. Using higher order derivatives of general nonlinear dynamics that need not be known, we construct a Koopman operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. The resulting error formula is used to choose the order of derivatives in the basis functions and obtain a data-driven Koopman model using a closed-form expression that can be computed in real time. Using the inverted pendulum system, we illustrate the robustness of the error bounds given noisy measurements of unknown dynamics, where the derivatives are estimated numerically. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods, such as SINDy and NARX. In addition, as a linear model, the Koopman approach lends itself readily to efficient control design tools, such as LQR, whereas the other modeling approaches require nonlinear control methods. The efficacy of the approach is further demonstrated with simulation and experimental results on the control of a tail-actuated robotic fish. Experimental results show that the proposed data-driven control approach outperforms a tuned PID (Proportional Integral Derivative) controller and that updating the data-driven model online significantly improves performance in the presence of unmodeled fluid disturbance. This paper is complemented with a video: https://youtu.be/9 wx0tdDta0. 

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2. Adaptive Robust Tracking Control for Hybrid Models of Three-Dimensional Bipedal Robotic Walking Under Uncertainties

   https://doi.org/10.1115/1.4050259  

   Gu, Yan; Yuan, Chengzhi (August 2021, Journal of Dynamic Systems, Measurement, and Control)

   Abstract This paper introduces an adaptive robust trajectory tracking controller design to provably realize stable bipedal robotic walking under parametric and unmodeled uncertainties. Deriving such a controller is challenging mainly because of the highly complex bipedal walking dynamics that are hybrid and involve nonlinear, uncontrolled state-triggered jumps. The main contribution of the study is the synthesis of a continuous-phase adaptive robust tracking control law for hybrid models of bipedal robotic walking by incorporating the construction of multiple Lyapunov functions into the control Lyapunov function. The evolution of the Lyapunov function across the state-triggered jumps is explicitly analyzed to construct sufficient conditions that guide the proposed control design for provably guaranteeing the stability and tracking the performance of the hybrid system in the presence of uncertainties. Simulation results on fully actuated bipedal robotic walking validate the effectiveness of the proposed approach in walking stabilization under uncertainties. 

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3. Fault Detection and Isolation of Uncertain Nonlinear Parabolic PDE Systems

   https://doi.org/https://doi.org/10.48550/arXiv.2203.15850  

   Zhang, Jingting; Yuan, Chengzhi; Zeng, Wei; Wang, Cong. (March 2022, ArXivorg)

   This paper proposes a novel fault detection and isolation (FDI) scheme for distributed parameter systems modeled by a class of parabolic partial differential equations (PDEs) with nonlinear uncertain dynamics. A key feature of the proposed FDI scheme is its capability of dealing with the effects of system uncertainties for accurate FDI. Specifically, an approximate ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the original PDE system. An adaptive dynamics identification approach using radial basis function neural network is then proposed based on this ODE system, so as to achieve locally-accurate identification of the uncertain system dynamics under normal and faulty modes. A bank of FDI estimators with associated adaptive thresholds are finally designed for real-time FDI decision making. Rigorous analysis on the FDI performance in terms of fault detectability and isolatability is provided. Simulation study on a representative transport-reaction process is conducted to demonstrate the effectiveness and advantage of the proposed approach. 

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4. Gaussian Control Barrier Functions: Safe Learning and Control

   https://doi.org/10.1109/CDC42340.2020.9303753  

   Khan, Mouhyemen; Chatterjee, Abhijit (December 2020, IEEE Conference on Decision and Control)

   null (Ed.)

   Safety is a critical component in today's autonomous and robotic systems. Many modern controllers endowed with notions of guaranteed safety properties rely on accurate mathematical models of these nonlinear dynamical systems. However, model uncertainty is always a persistent challenge weakening theoretical guarantees and compromising safety. For safety-critical systems, this is an even bigger challenge. Typically, safety is ensured by constraining the system states within a safe constraint set defined a priori by relying on the model of the system. A popular approach is to use Control Barrier Functions (CBFs) that encode safety using a smooth function. However, CBFs fail in the presence of model uncertainties. Moreover, an inaccurate model can either lead to incorrect notions of safety or worse, incur system critical failures. Addressing these drawbacks, we present a novel safety formulation that leverages properties of CBFs and positive definite kernels to design Gaussian CBFs. The underlying kernels are updated online by learning the unmodeled dynamics using Gaussian Processes (GPs). While CBFs guarantee forward invariance, the hyperparameters estimated using GPs update the kernel online and thereby adjust the relative notion of safety. We demonstrate our proposed technique on a safety-critical quadrotor on SO(3) in the presence of model uncertainty in simulation. With the kernel update performed online, safety is preserved for the system. 

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5. Stable nullspace adaptive parameter identification of 6 degree-of-freedom plant and actuator models for underactuated vehicles: Theory and experimental evaluation

   https://doi.org/10.1177/02783649231191184  

   Harris, Zachary J.; Mao, Annie M.; Paine, Tyler M.; Whitcomb, Louis L. (November 2023, The International Journal of Robotics Research)

   Model-based approaches to navigation, control, and fault detection that utilize precise nonlinear models of vehicle plant dynamics will enable more accurate control and navigation, assured autonomy, and more complex missions for such vehicles. This paper reports novel theoretical and experimental results addressing the problem of parameter estimation of plant and actuator models for underactuated underwater vehicles operating in 6 degrees-of-freedom (DOF) whose dynamics are modeled by finite-dimensional Newton-Euler equations. This paper reports the first theoretical approach and experimental validation to identify simultaneously plant-model parameters (parameters such as mass, added mass, hydrodynamic drag, and buoyancy) and control-actuator parameters (control-surface models and thruster models) in 6-DOF. Most previously reported studies on parameter identification assume that the control-actuator parameters are known a priori. Moreover, this paper reports the first proof of convergence of the parameter estimates to the true set of parameters for this class of vehicles under a persistence of excitation condition. The reported adaptive identification (AID) algorithm does not require instrumentation of 6-DOF vehicle acceleration, which is required by conventional approaches to parameter estimation such as least squares. Additionally, the reported AID algorithm is applicable under any arbitrary open-loop or closed-loop control law. We report simulation and experimental results for identifying the plant-model and control-actuator parameters for an L3 OceanServer Iver3 autonomous underwater vehicle. We believe this general approach to AID could be extended to apply to other classes of machines and other classes of marine, land, aerial, and space vehicles. 

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